831 Characterization of subglacial landscapes by a two-parameter roughness index Xin LI,1,2 Bo SUN,1 Martin J. SIEGERT,3 Robert G. BINGHAM,4 Xueyuan TANG,1 Dong ZHANG,1 Xiangbin CUI,1 Xiangpei ZHANG1 1Polar Research Institute of China, 451 Jinqiao Road, Pudong, Shanghai 200136, China E-mail: [email protected] 2School of Ocean and Earth Sciences, Tongji University, Shanghai 200092, China 3School of GeoSciences, University of Edinburgh, King’s Buildings, Edinburgh EH9 3JW, UK 4School of Geosciences, University of Aberdeen, Elphinstone Road, Aberdeen AB24 3UF, UK ABSTRACT. Previous studies using Fourier transformation (FT) methods to analyze subglacial roughness have shown promise for distinguishing between different types of subglacial landscape from raw subglacial elevation data. We derive a two-parameter FT roughness index {, }, where is based on the FT of elevation (as previously considered in isolation), and is based on both the FT of elevation and the FT of bed-slope profile. In this way, we take account of both vertical and horizontal irregularities in subglacial surfaces. We demonstrate the statistical veracity of using {, } to consider roughness in terms of obstacle amplitudes and spacing, and consider the use of {, } in studies of ice dynamics and subglacial geomorphological interpretation. We show that {, } can be linked to basal sliding rates on the metre scale, and can be used to differentiate further than single-parameter roughness indices between different classes of subglacial landscape, in particular between erosional and depositional settings. 1. INTRODUCTION described in this paper, have made use of the concept of Statistically characterizing the topographic structure of the subglacial ‘roughness’. basal boundary beneath ice masses is increasingly being Generally, roughness is regarded as the irregularity of a recognized as an important imperative in glaciological surface. In glaciology, the initial attempt at expressing research (e.g. Hubbard and Hubbard, 1998; Taylor and subglacial bed roughness was due to the requirement of others, 2004; Rippin and others, 2006). Subglacial topog- estimating how a glacier’s basal sliding is controlled by the raphy at all scales imparts important clues concerning the undulation of bedrock. Weertman (1957) considered an distribution of subglacial sediments, subglacial geomorphic idealized case of an undulating bed as cubic obstacles with processes and ice-stream stability (Hubbard and others, length, a, distributed periodically with a separation distance, 2000; Bingham and Siegert, 2009), and is thought to exert a , between each obstacle. Put another way, the two fundamental control on ice dynamics (Weertman, 1957; parameters, a and , in this model denote the amplitude Siegert and others, 2004, 2005). Nevertheless, to date, few and spatial frequency of the undulations of the idealized studies have examined subglacial topographic patterns and bed. Expanding Weertman’s theory to consider bedrock their influences quantitatively, perhaps due to the paucity, surfaces with randomly distributed undulations, Kamb until recently, of appropriate bed profile datasets from (1970) and Nye (1970) introduced Fourier transformations contemporary glacierized environments. Modern techniques (FTs) of bed elevation profiles into their theories. A FT can be have alleviated this issue: radio-echo sounding (RES) has used to transform any surface into a sum of several enabled the acquisition of myriads of bed profiles at the periodically corrugated surfaces, and can thus be used to regional scale (decimetres to kilometres) across ice sheets express the amplitude and spatial frequency of the range of and ice caps (e.g. Lythe and others 2001; Gogineni and undulations present. Hence, these and follow-up studies others, 2007; Sun and others, 2009), Hubbard and others have quantified roughness in terms of two key parameters: (2000) have demonstrated the ease with which centimetre- the vertical irregularity (amplitude) and the horizontal and metre-scale bed profiles may be obtained across irregularity (spatial frequency); or in an equivalent way, as recently deglaciated terrain using microroughness meters slope is the bridge between vertical and horizontal, using and electro-optical distance meters respectively, and terres- amplitude and slope (Paterson, 1994). trial laser scanning systems have shown great potential in Though characterizing roughness by FT works well in surface characterization, especially with respect to three- theoretical dynamic studies as described above, it is not dimensional (3-D) surface characterization (e.g. Bauer and convenient for showing the spatial distribution of roughness. others, 2003). As ‘real’ bed data have thus become easier to A basic tenet of cartography is that showing the spatial obtain, and there is an increasing drive to integrate such data distribution of any property ideally requires that it is into numerical models of ice dynamics and ice-stream displayed in terms of spatial variations in the magnitude of behaviour (Solomon and others, 2007; Pattyn and others, a single parameter. Unfortunately, a full Fourier spectrum for 2008; Hindmarsh, 2009), the need for characterizing sub- roughness is too complicated to meet this requirement. glacial topography in a statistically meaningful manner has Hubbard and others (2000) and Taylor and others (2004) arisen. In this context, recent studies, and the work circumvented this problem by defining a single-parameter 832 Li and others: Subglacial landscape characterization by FT roughness index Fig. 1. Random surfaces of different roughness, generated using a self-correlation function (Thomas, 1999). (a–d) show the raw elevation profiles, and (e–h) their respective spectral power densities. All { , } are in units of { b, b}. roughness index defined as the integral of the spectrum subglacial roughness across Antarctica in several previous within a specified wavelength interval; this method has since studies (e.g. Siegert and others, 2004; Bingham and Siegert, been applied to show the spatial distribution of subglacial 2009). Figure 1 illustrates this method for four rough roughness across Antarctica (Siegert and others, 2004, 2005; surfaces. The surface in Figure 1a is clearly less rough than Bingham and Siegert, 2007, 2009). In this paper, we build that in Figure 1b, and this is reflected in their t values (0.3 on this method by additionally considering the FT of bed- and 1.0 respectively). However, while the surfaces shown in slope profiles as well as that (traditionally considered) of Figure 1b–d all yield the same t values, they are bedrock amplitude, and link it to horizontal irregularities. qualitatively quite different from one another: the surface We believe that calculating the FT of both bed slope and in Figure 1b is steeper than that in Figure 1d, and that in amplitude enhances the robustness of spectral roughness Figure 1c is the steepest. We suggest, therefore, that while techniques in subglacial landscape characterization and the parameter has demonstrably been useful for describing process inference. roughness, in isolation it may not be sufficient. Here we investigate the utility of considering the extra factor, the FT of bed-slope profile. 2. METHOD Returning to our conception of roughness in terms of We begin by calculating the single-parameter roughness vertical irregularities (amplitude) and horizontal irregular- index of a profile of length, (–l/2,l/2). We first remove the ities (spatial frequency), we can interpret Figure 1b–d as mean, hiZðxÞ from the bed elevation profile, Z(x): follows: the amplitudes of the undulations are similar, but the frequencies of the undulations differ. Hence, a factor Z ðxÞ¼ZðxÞhiZðxÞ , ð1Þ 0 related to horizontal irregularity needs to be introduced. where x corresponds to the horizontal location and Z Given that a bed-slope profile expresses the connection corresponds to the raw elevation data. We then apply a FT to between the vertical and horizontal dimensions, it is obtain the spectral power density, S(k): reasonable to choose some statistical function of slope as the extra factor. A slope profile, sl(x), can be calculated from ð Þ¼1 ð Þ 2 ð Þ S k Z 0 k , 2 the elevation profile, Z (x), by sl(x)=@Z (x)/@x, and its l 0 0 spectral power density, S (k), is ð Þ sl where Z0 k is the FT of Z0(x). The roughness index, ,is 1 2 obtained by integrating S(k) in a specified wavelength S ðkÞ¼ slðkÞ : ð4Þ sl l interval, (k1,k2): Z k2 Following Equation (4), we obtain the slope index, sl,by ¼ ð Þ : ð Þ Z S k dk 3 k2 k1 ¼ ð Þ : ð Þ sl Ssl k dk 5 k1 When k1 is zero and k2 is infinity, we obtain a ‘total roughness index’, t. This is the property used to characterize Finally, as slope is the ratio of vertical dimension to Li and others: Subglacial landscape characterization by FT roughness index 833 horizontal dimension, we define the parameter , corres- ponding to horizontal irregularity alone and invariant of vertical stretching of the profile, as R k2 SðkÞ dk ¼ ¼ R k1 k2 ð Þ sl Ssl k dk k1 ð Þ R ÀÁR 1 R 1 0 6 k2 1 Àikx 0 ikx 0 1 Z0ðxÞe dx 1 Z0ðx Þe dx dk ¼ Rk1 l R R : 1 ð Þ 1 ð 0Þ 0 k2 1 dZ0 x Àikx dZ0 x ikx 0 e dx 0 e dx dk k1 l 1 dx 1 dx Our two-parameter roughness index, taking into account both the vertical and the horizontal irregularities, is thus defined as {, }. Thus, while describes the magnitude of vertical deviations in the bed slope, quantifies the horizontal frequency of these deviations. This distinction is expressed graphically in Figure 1 (and is illuminated mathematically in section 3). Here each surface in Fig. 2. (a) 220 km subglacial elevation profile from East Ant- Figure 1b–d, which give the same value of t, yields different arctica, acquired by 21st CHINARE (Sun and others, 2009). (b, c) values of t. Figure 1a and b, which have the same t,have Corresponding distribution of and respectively.